Least squares null space variational characterization for nonminimum norm solutions

The least-squares estimation problem with non-minimum-norm constraints on the unknown model parameters is considered. Contrary to the quadratic-constraint least-squares solutions the approach presented does not necessarily satisfy the constraint, but rather relies on the nullity of the data matrix to maintain the unconstrained least-squares error value while trading off the minimum-norm solution by another with the shortest distance from the null space of the constraint. The singular value decomposition of the data matrix is used to obtain the necessary information about the minimum-norm solution as well as the basis of the null space. Closed-form expressions are derived for the case in which the constraint of interest is the smoothness of the model parameters. Examples of sinusoids in white noise are given for illustration